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Register a new userFinite-Dimensional Spin Glasses: States, Excitations, and Interfaces
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C.M. Newman
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We discuss the underlying connections among the thermodynamic properties of short-ranged spin glasses, their behavior in large finite volumes, and the interfaces that separate different pure states, and also ground states and low-lying excitations.
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We present a general theorem restricting properties of interfaces between thermodynamic states and apply it to the spin glass excitations observed numerically by Krzakala-Martin and Palassini-Young in spatial dimensions d=3 and 4. We show that such excitations, with interface dimension smaller than d, cannot yield regionally congruent thermodynamic states. More generally, zero density interfaces of translation-covariant excitations cannot be pinned (by the disorder) in any d but rather must deflect to infinity in the thermodynamic limit. Additional consequences concerning regional congruence in spin glasses and other systems are discussed.
We use a non-equilibrium simulation method to study the spin glass transition in three-dimensional Ising spin glasses. The transition point is repeatedly approached at finite velocity $v$ (temperature change versus time) in Monte Carlo simulations starting at a high temperature. The normally problematic critical slowing-down is not hampering this kind of approach, since the system equilibrates quickly at the initial temperature and the slowing-down is merely reflected in the dynamic scaling of the non-equilibrium order parameter with $v$ and the system size. The equilibrium limit does not have to be reached. For the dynamic exponent we obtain $z = 5.85(9)$ for bimodal couplings distribution and $z=6.00(10)$ for the Gaussian case, thus supporting universal dynamic scaling (in contrast to recent claims of non-universal behavior).
We present a detailed proof of a previously announced result (C.M. Newman and D.L. Stein, Phys. Rev. Lett. v. 84, pp. 3966--3969 (2000)) supporting the absence of multiple (incongruent) ground state pairs for 2D Edwards-Anderson spin glasses (with zero external field and, e.g., Gaussian couplings): if two ground state pairs (chosen from metastates with, e.g., periodic boundary conditions) on the infinite square lattice are distinct, then the dual bonds where they differ form a single doubly-infinite, positive-density domain wall. It is an open problem to prove that such a situation cannot occur (or else to show --- much less likely in our opinion --- that it indeed does happen) in these models. Our proof involves an analysis of how (infinite-volume) ground states change as (finitely many) couplings vary, which leads us to a notion of zero-temperature excitation metastates, that may be of independent interest.
We investigate the performance of the recently proposed stationary Fokker-Planck sampling method considering a combinatorial optimization problem from statistical physics. The algorithmic procedure relies upon the numerical solution of a linear second order differential equation that depends on a diffusion-like parameter D. We apply it to the problem of finding ground states of 2d Ising spin glasses for the +-J-Model. We consider square lattices with side length up to L=24 with two different types of boundary conditions and compare the results to those obtained by exact methods. A particular value of D is found that yields an optimal performance of the algorithm. We compare this optimal value of D to a percolation transition, which occurs when studying the connected clusters of spins flipped by the algorithm. Nevertheless, even for moderate lattice sizes, the algorithm has more and more problems to find the exact ground states. This means that the approach, at least in its standard form, seems to be inferior to other approaches like parallel tempering.
We present a mean field model for spin glasses with a natural notion of distance built in, namely, the Edwards-Anderson model on the diluted D-dimensional unit hypercube in the limit of large D. We show that finite D effects are strongly dependent on the connectivity, being much smaller for a fixed coordination number. We solve the non trivial problem of generating these lattices. Afterwards, we numerically study the nonequilibrium dynamics of the mean field spin glass. Our three main findings are: (i) the dynamics is ruled by an infinite number of time-sectors, (ii) the aging dynamics consists on the growth of coherent domains with a non vanishing surface-volume ratio, and (iii) the propagator in Fourier space follows the p^4 law. We study as well finite D effects in the nonequilibrium dynamics, finding that a naive finite size scaling ansatz works surprisingly well.
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